18 research outputs found
Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes
This article exhibits a 4-dimensional combinatorial polytope that has no
antiprism, answering a question posed by Bernt Lindst\"om. As a consequence,
any realization of this combinatorial polytope has a face that it cannot rest
upon without toppling over. To this end, we provide a general method for
solving a broad class of realizability problems. Specifically, we show that for
any semialgebraic property that faces inherit, the given property holds for
some realization of every combinatorial polytope if and only if the property
holds from some projective copy of every polytope. The proof uses the following
result by Below. Given any polytope with vertices having algebraic coordinates,
there is a combinatorial "stamp" polytope with a specified face that is
projectively equivalent to the given polytope in all realizations. Here we
construct a new stamp polytope that is closely related to Richter-Gebert's
proof of universality for 4-dimensional polytopes, and we generalize several
tools from that proof
Transversals and colorings of simplicial spheres
Motivated from the surrounding property of a point set in
introduced by Holmsen, Pach and Tverberg, we consider the transversal number
and chromatic number of a simplicial sphere. As an attempt to give a lower
bound for the maximum transversal ratio of simplicial -spheres, we provide
two infinite constructions. The first construction gives infintely many
-dimensional simplicial polytopes with the transversal ratio exactly
for every . In the case of , this meets the
previously well-known upper bound tightly. The second gives infinitely
many simplicial 3-spheres with the transversal ratio greater than . This
was unexpected from what was previously known about the surrounding property.
Moreover, we show that, for , the facet hypergraph
of a -dimensional simplicial sphere
has the chromatic number , where is the number of vertices of . This
slightly improves the upper bound previously obtained by Heise, Panagiotou,
Pikhurko, and Taraz.Comment: 22 pages, 2 figure
Realization spaces of arrangements of convex bodies
We introduce combinatorial types of arrangements of convex bodies, extending
order types of point sets to arrangements of convex bodies, and study their
realization spaces. Our main results witness a trade-off between the
combinatorial complexity of the bodies and the topological complexity of their
realization space. First, we show that every combinatorial type is realizable
and its realization space is contractible under mild assumptions. Second, we
prove a universality theorem that says the restriction of the realization space
to arrangements polygons with a bounded number of vertices can have the
homotopy type of any primary semialgebraic set
Completeness for the Complexity Class ∀ ∃ R and Area-Universality
Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class ∃R plays a crucial role in the study of geometric problems. Sometimes ∃R is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, ∃R deals with existentially quantified real variables. In analogy to Πp2 and Σp2 in the famous polynomial hierarchy, we study the complexity classes ∀∃R and ∃∀R with real variables. Our main interest is the AREA UNIVERSALITY problem, where we are given a plane graph G, and ask if for each assignment of areas to the inner faces of G, there exists a straight-line drawing of G realizing the assigned areas. We conjecture that AREA UNIVERSALITY is ∀∃R -complete and support this conjecture by proving ∃R - and ∀∃R -completeness of two variants of AREA UNIVERSALITY. To this end, we introduce tools to prove ∀∃R -hardness and membership. Finally, we present geometric problems as candidates for ∀∃R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability